A right triangle is the easiest triangle to acquire the area for using the basic formula A=1/2 base * height. However, you can't use that formula for the non-right triangles you might encounter, so the other two formulas are a must for finding the area of ALL general triangles!
the 3 formulas
Heron's formula is used when given the 3 side lengths but no angles and can be used on both right and oblique triangles. The formula is Area= s(s−a)(s−b)(s−c) where s=(a+b+c)2. The kindergarten or basic formula is Area= 1/2 base * height but can only be used on right triangles. The general formula can be used on any triangle with two side lengths and their included angle. Area= (the side lengths can be ANY two side lengths as long as the angle is their included angle.) It is important to know multiple area formulas for right triangles because you won't necessarily be given the same information for each triangle. For example, if you are given side lengths of 3, 4, and 5 you would be able to plug into your formula area= 6(6−3)(6−4)(6−5) and you should get an area of 6. If you are given a right triangle with a height of 4 and a base of 3 you would be able to use the kindergarten formula area= 1/2(3)(4) and you should get area=6. If you are given side length a=4 and side length b=5 and angle C=30 degrees you can use the general formula area= 1/2(4)(5)sin(30) your area should be 5.
which formula when?
You should use the Law of Cosines when you have determined that you can't use the Law of Sines and have two side lengths and their included angle OR all three side lengths. If you are given all three sides use the formula cosA=(a2−b2−c2)−2bc. You will then have a completed ratio for your A angle and side and can use the Law of Sine to solve for the other two angles. If you are given two sides and their included angles use the formula a2=b2+c2−2(b)(c)cosA. You again will have a completed ratio for your A angle and side and can use law of sine to solve for the other two angles.
You would use the Law of Sines when given an angle and its corresponding side lengths. It is most effective when given another side length or angle. When solving the triangle you may end up with 1, 2, or no triangle solutions, depending on how much information is given and if the given information is plausible. For example, if you are only given angle A=30 degrees, side a=5, and side b=23 you would solve for b using a portion of the Law of Sines formula, Sin30/5=SinB/23; you would get an angle of 2.3 degrees and assume another solution of 187.7 degrees. Because the second solution was an impossible solution for a triangle you would only use the first answer when solving for angle C and side c. However, had the situation resulted in two possible B values you would use both to solve for C and c and write all the solutions for both triangles. If you have an error when solving for your B or b value, you will have no solution as the information given can not create a triangle.
You would use the Law of Sines when given an angle and its corresponding side lengths. It is most effective when given another side length or angle. When solving the triangle you may end up with 1, 2, or no triangle solutions, depending on how much information is given and if the given information is plausible. For example, if you are only given angle A=30 degrees, side a=5, and side b=23 you would solve for b using a portion of the Law of Sines formula, Sin30/5=SinB/23; you would get an angle of 2.3 degrees and assume another solution of 187.7 degrees. Because the second solution was an impossible solution for a triangle you would only use the first answer when solving for angle C and side c. However, had the situation resulted in two possible B values you would use both to solve for C and c and write all the solutions for both triangles. If you have an error when solving for your B or b value, you will have no solution as the information given can not create a triangle.
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This proof for the law of cosine allows me to make sure the law is true for all triangles. The law of sines was more obvious as it dealt with ratios for which I am quite familiar, but the law of cosines wasn't so transparent so the proof helped to verify and clarify!
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